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32 {\LARGE {\bf Materials Physics I}\\}
37 {\Large\bf Tutorial 2}
43 \item $r_i=r_{i0}+u_i$\\
44 $\rho=r_2-r_1=r_{20}+u_2-r_{10}-u_1=(r_{20}-r_{10})+(u_2-u_1)
46 \item $\Phi-\Phi_0=\frac{D}{2}(\rho-\rho_0)^2
47 =\frac{D}{2}(\rho^2+\rho_0^2-2\rho_0\rho)$\\
48 $\rho^2=\rho_0^2+\sigma^2+2\rho_0\sigma$
49 $\Rightarrow$ $\rho=\sqrt{\rho_0^2+\sigma^2+2\rho_0\sigma}$\\
50 $\Rightarrow$ $\Phi-\Phi_0=\frac{D}{2}
51 [2\rho_0^2+\sigma^2+2\rho_0\sigma-
52 2\rho_0\sqrt{\rho_0^2+\sigma^2+2\rho_0\sigma}]$
54 \item $\sigma \parallel \rho_0$:
56 \item \begin{flushleft}
57 \includegraphics[height=6cm]{elongation_p01.eps}
58 \includegraphics[height=6cm]{elongation_p02.eps}
59 \includegraphics[height=6cm]{elongation_p03.eps}
61 \item $\sigma = \sigma_{\parallel}$:\\
62 $\rho_0 \sigma_{\parallel} = |\rho_0| |\sigma_{\parallel}|$\\
63 $\Phi-\Phi_0=\frac{D}{2}\left(2\rho_0^2+\sigma_{\parallel}^2+
64 2\rho_0\sigma_{\parallel}-
65 2\rho_0\sqrt{(\rho_0+\sigma_{\parallel})^2}\right)
66 =\frac{D}{2}\sigma_{\parallel}^2$
68 \item $\sigma \perp \sigma_0$:
70 \item \begin{flushleft}
71 \includegraphics[height=5.3cm]{elongation_n01.eps}
72 \includegraphics[height=5.3cm]{elongation_n02.eps}
73 \includegraphics[height=5.3cm]{elongation_n03.eps}
75 \item $\sigma=\sigma_{\perp}$:\\
76 $\sigma_{\perp} \rho_0 = 0$\\
77 $\Phi-\Phi_0=\frac{D}{2}\left[2\rho_0^2+\sigma_{\perp}^2-
78 2\rho_0\sqrt{\rho_0^2+\sigma_{\perp}^2}\right]$
80 \item $\sigma_{\perp} = \alpha \rho_0$, $\alpha \ll 1$\\
81 $\sqrt{\rho_0^2+\sigma_{\perp}^2}=
82 \sqrt{\rho_0^2+\alpha^2\rho_0^2}=
83 \rho_0\sqrt{1+\alpha^2}=
84 \rho_0(1+\frac{\alpha^2}{2}-\frac{\alpha^4}{8}+\ldots)$\\
85 $\Rightarrow \Phi-\Phi_0=
86 \frac{D}{2}\left[\rho_0^2\left(2+\alpha^2-
87 2(1+\frac{\alpha^2}{2}-\frac{\alpha^4}{8}+\ldots)\right)\right]=
88 \frac{D}{2}\left[\rho_0^2(\frac{\alpha^4}{4}+\ldots)\right]$\\
89 $\Rightarrow \Phi-\Phi_0\stackrel{\alpha\ll 1}{=}
90 \frac{D}{2}\rho_0^2\frac{\alpha^4}{4}=
91 \frac{D}{2}\sigma_{\perp}^2\frac{\alpha^2}{4}$
92 \item $\sigma_{\parallel}$, $\sigma_{\perp} \ll \rho_0$\\
93 $\Rightarrow$ potential contribution of $\sigma_{\perp}$
94 compared to contribution of $\sigma_{\parallel}$
98 \item As long as the displacements and thus the elongation is small
99 compared to the equilibrium state the change in the potential
100 due to the perpendicular elongation is negligible small.
101 \item Regarding a possible existence of perpendicular elongation
102 the model of the linear chain is unproblematic.
103 \item In a real crystal couplings in other directions exist.
104 These can only be neglected if they are small compared to the
105 coupling of the considered direction.
111 \item Derive the dispersion relation for a linear chain with two different
112 alternating types of atoms.
113 \item Discuss the two solutions for $\omega^2$.